# Imaginary. Axis. Real. Axis

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1 Name ME6 Final. I certify that I upheld the Stanford Honor code during this exam Monday December 2, 25 3:3-6:3 p.m. ffl Print your name and sign the honor code statement ffl You may use your course notes, homework, books, etc. ffl Write your answers on this handout ffl Where space is provided, show your work to get credit ffl If necessary, attach extra pages for scratch work ffl Best wishes for a fun vacation. Merry Christmas and Happy New Year! Page Value Score Total F. (2 pts.) Miscellaneous (a) ( pt.) What is your Meyer-Briggs (human metrics) personality type? (b) ( pt.) Suppose you plan on teaching or TAing Dynamic Systems. Provide suggestions for improving the course. Specifically, what you would add, remove, or change in the course (e.g., content, homework, book, lectures, etc.)?

2 F.2 (3 pts.) Coupled motions of Wilbur-force pendulum N N o Shown to the right is a rigid body B that is attached to a spring at B c, the mass center of B. The other end of the spring is attached to point N o which is fixed in a Newtonian reference frame N. The equations governing this model of the system are B m ẍ + k x x + k c = I + k c x + k = B c Quantity Symbol Type Mass of B m constant Central moment of inertia of B about vertical I constant Linear spring constant modeling extensional flexibility k x constant Linear spring constant modeling torsional flexibility k constant Linear spring constant modeling coupled flexibility k c constant Translational stretch of spring from equilibrium x dependent variable Rotational stretch of spring from equilibrium dependent variable (a) (2 pts.) Write the ODEs in the matrix form M Ẍ + B _X + KX =, where X =» x» m I» ẍ +»» _x _ + " k x k c k c k #» x (b) (9 pts.) The solution to the previous set of ODEs has the form X(t)=U Λe pt where p is a constant (to-be-determined) and U is a non-zero 2 matrix of constants (to-be-determined). Fill in the two blanks in the following polynomial equation that governs the values of = - p 2. Note: the blanks only involve m, I, k x, k, k c. =» kx m + -k I Λ + k x k k 2 c mi = 2

3 (c) (6 pts.) For certain values of m, I, k x, k, and k c, the matrix A = M - K = Find the eigenvalues and corresponding eigenvectors of A.». = 9 U =» - 2 = U 2 =» (d) (2 pts.) Calculate the values of p, p 2, p 3, p 4 and draw their locations in the complex plane. p ;2 = ± p - = ± p -9 = ± 3 i p 3;4 = ± p - 2 = ± p - = ± 3:32 i Imaginary Axis x x Real Axis (e) ( pt.) The solution is stable/ neutrally stable /unstable. (f) (4 pts.) Write the solution for X(t) in terms of the yet-to-be-determined constants c, c 2, c 3, c 4, the sine and cosine functions, and t.» x(t) (t) =» - n c sin( 3 t) + c 2 cos( 3 t)o 3 +» n c 3 sin( 3:32 t) + c 4 cos( 3:32 t)o

4 (g) (2 pts.) Determine c, c 2, c 3, c 4 when x() = :2, ()=, _x() =, and _ () =. c = c 2 = -: c 3 = c 4 = : (h) (2 pts.) Using the aforementioned initial values, write explicit solutions for x(t) and (t). x(t) = : cos(3 t) + : cos(3:32 t) (t) = -: cos(3 t) + : cos(3:32 t) (i) ( pt.) Using trigonometric identities, equation (2.9), and cos(a) =-cos(a+ß), show that your previous solution for x(t) and (t) is x(t) = :2 sin(-:6 t + ß 2 ) sin(3:6 t + ß 2 ) = :2 cos(:6 t) cos(3:6 t).2.2 (t) = :2 sin(-:6 t) sin(3:6 t) = :2 sin(:6 t + ß) sin(3:6 t).5.5 Displacement Rotation angle (radians) (j) (2 pts.) Give an interpretation of the time-behavior of x(t) and (t). Both x(t)and (t)exhibit the beat phenomena with a high-frequency of 3:6 rad sec rad and a low-frequency of :6 sec. Since x(t)and (t)are coupled, x's maximum amplitude coincides with 's minimum amplitude (and vice-versa). 4

5 F.3 (38 pts.) Linearization of equations of motion for a swinging spring A straight, massless, linear spring connects a particle Q to a point N o which is fixed in a Newtonian reference frame N. The system identifiers and governing equations are shown below. Quantity Identifier Type Local gravitational constant g constant Mass of Q m constant Natural spring length L n constant Linear spring constant k constant Spring stretch y variable Pendulum angle" variable m (L n + y) + 2 m _ _y + mg sin( ) = m ÿ m (L n +y) _ 2 + ky mg cos( ) = N N o b y k θ L n n z n y n x y b x (a) ( pt.) Classify the previous equations by picking the relevant qualifiers from the following list. Uncoupled Linear Homogeneous Constant-coefficient st-order Algebraic Coupled Nonlinear Inhomogeneous Variable-coefficient 2nd-order Differential (b) (3 pts.) Find the condition on y nom such that y = y nom (a constant) and = nom = satisfy the nonlinear ODEs. y nom = mg k (c) ( pts.) Using a Taylor series, linearize the differential equations in perturbations about this nominal solution. In other words, define perturbations of y and as ey = y y nom (y nom is constant) e = nom ( nom =) and form a set of differential equations which arelineariney, _ ey, ëy, e, _ e, ë. m (L n + y nom ) ë + mg e = 5 m ëy + k ey =

6 (d) (7 pts.) Replace y and in the original nonlinear ODEs with y = ey + y nom (y nom is the constant determined earlier) = e + nom ( nom =) Assuming ey, _ ey, ëy and e, _ e, and ë are small and using the small-approximations technique of Chapter 2, form a set of ODEs that are linear in ey, _ ey, ëy, e, _ e, ë. m (L n + y nom ) ë + mg e = m ëy + k ey = (e) ( pt.) Classify your equations in part (3d) by picking the relevant qualifiers from the following list. Uncoupled Linear Homogeneous Constant-coefficient st-order Algebraic Coupled Nonlinear Inhomogeneous Variable-coefficient 2nd-order Differential (f) ( pt.) The linearized ODEs resulting from the Taylor series technique are identical to those obtained from the small-approximations technique. True /False. (g) (5 pts.) Find analytical solutions for ey(t) and (t) e when g =9:8 m=sec 2, m = kg, L n =:5 m, k = n=m, ey() = : m, e () = ffi,and _ ey() = _ e () = ey(t) = e (t) = q k : cos( t) m = : cos( t) q q ffi g cos( L n+y t) nom = ffi cos( gk t) = L n k+mg ffi cos(4:5 t) 6

7 (h) (4 pts.) Make a rough sketch of your linearized solutions for y(t)and (t)for» t» 6 sec, showing the relevant characteristics, e.g., amplitude, frequency, growth, decay, etc. y(t) ß ey(t)+y nom = : cos( t)+:98 (t) ß e (t)+ nom = ffi cos(4:5 t).2 3 Spring stretch (meters).5..5 Pendulum angle (degrees) (i) (6 pts.) The computer solution for y(t) and (t) to the original nonlinear ODEs for» t» 6 sec are plotted below. Comment on the similarities and differences between the plots you produced with the ones below. Give a reason for the similarities and differences..2 3 Spring stretch (meters) Similarities: Frequency, maximum amplitude Pendulum angle (degrees) Similarities: Frequency Differences: Beat phenomenon, varying amplitude Differences: Amplitude, beat phenomenon Reason for differences: Coupled and nonlinear Reason for differences: Coupled and nonlinear 7

8 F.4 (29 pts.) Dynamic response for an air-conditioner on a building An air conditioner is bolted to the roof of a one story building. The air conditioner's motor is unbalanced and its eccentricity is modeled as a particle of mass m attached to the distal end of a rigid rod of length r. When the motor spins with angular speed Ω, it causes the building's roof of mass M to vibrate. The stiffness and material damping in each column that supports the roof is modeled as a linear horizontal spring (k) and linear horizontal damper (b). Homework 4.5 showed that the ODE governing the horizontal displacement x of the building's roof is (M +m)ẍ + 2 b _x + 2 kx = mrω 2 sin(ωt) (a) (3 pts.) Express! n,, A in terms of M, m, b, k, r so the ODE is x k,b M Ωt r m k,b ẍ + 2! n _x +! 2 n x = AΩ2 sin(ωt)! n = q 2 k b = p M + m (M+m) k A = mr M + m (b) (2 pts.) For certain values of M, m, b, k, andr, this ODE simplifies to Calculate numerical values for! n and. ẍ + 3 _x + 9 x = x -4 Ω 2 sin(ωt)! n = 3 rad sec = :5 (c) (7 pts.) Fillinnumerical valuesinthefollowing expressions for x ss (t), the steady-state part of x(t). Ω ( rad sec ) x ss(t) 2 7:94 x -5 Λ sin( 2 t + -6:843 ffi ) 3 : Λ sin( 3 t + -9 ffi ) 4 2:25 x -4 Λ sin( 4 t + -7:3 ffi ) 8

9 (d) (4 pts.) The building's occupants complain that the roof shakes too much. Comment on Ω the effect small variations of M, m, b, k, and r have on:,! n, and the magnitude of x ss (t). Fill in each element in the table by writing (decreases), (no effect), + (increases), or? (if it may decrease or increase). For the 2 nd -to-last column, assume the air conditioner's normal operating speed is Ω = 28 rad rad, and for last column, assume Ω = 32. sec sec Ω!n Ω ß 28 rad sec jx ss (t)j Ω ß 32 rad sec jx ss (t)j Balancing the motor (r! ) Increasing the motor speed Ω + + Decreasing the motor speed Ω + Adding mass to the roof (increasing M) + +?? Removing mass from the roof +?? Stiffening the support columns (increasing k) + Adding damping to the columns (increasing b) + (e) (3 pts.) List three ways to change the motor and minimize the roof shaking. Balance the motor, damp the motor, or operate Ω fl! n or Ω fi! n. 9

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